Indirect NRDF for Partially Observable Gauss-Markov Processes with MSE Distortion: Complete Characterizations and Optimal Solutions

Abstract

<div>We develop a new sequential rate distortion function</div><div>to compute lower bounds on the average length of all causal prefix free codes for partially observable multivariate Markov processes with mean-squared error distortion constraint. Our information measure is characterized by a variant of causally conditioned directed information and is utilized in various application examples. First, it is used to optimally characterize a finite dimensional optimization problem for jointly Gaussian processes and to obtain the corresponding optimal linear encoding and decoding policies.</div><div>Under the assumption that all matrices commute by pairs,</div><div>we show that our problem can be cast as a convex program</div><div>which achieves its global minimum. We also derive sufficient</div><div>conditions which ensure that our assumption holds. We then</div><div>solve the KKT conditions and derive a new reverse-waterfilling algorithm that we implement. If our assumption is violated, one can still use our approach to derive sub-optimal (upper bound) waterfilling solutions. For scalar-valued Gauss-Markov processes with additional observation noise, we derive a new closed form solution and we compare it with known results in the literature. For partially observable time-invariant Markov processes driven by additive i:i:d: system noise only, we recover using an alternative approach and thus strengthening a recent result by Kostina and Hassibi in [1, Theorem 9] whereas for timeinvariant and spatially IID Markov processes driven by additive noise process we also derive new analytical lower bounds.</div>

State-Dependent Generalizations of Nonanticipatory Epsilon Entropy of Partially Observable Processes

“…Our results for (Q1) compliment recent developments in [18], where Gaussian processes (X n , Y n ) are considered, in the sense that we additionally derive recursive formulas of test-channel distributions for arbitrary processes (X n , Y n ).…”

“…Nevertheless, we show that the claims of [13,Lemma 2] and [18,Lemma 3] are valid, due to a structural property of the test-channel of Theorem 3.…”

“…Now, we turn to Question (Q5), i.e., we consider the jointly Gaussian partially observable process 2 . We emphasize that the analysis of this section, compliments [13] and the recent paper [18], in the following aspects: 1) An optimal test-channel Lemma 2] that depends on y n through E{X t X t }, but no derivation is given to support this claim, and no realization of X n is given. 2) In [18,Lemma 3] it is proven that the optimal test-channel…”

“…these degenerate to [23,Proposition 3]. Theorem 1 and [23, Proposition 3], are fundamentally different from related bounds in [13], [14], [18] that make causal use of past reproductions S n = X n−1 at the encoder (which do not use the state variable Z n ). Theorem 2 (in Section III-A), gives conditions for the lower bounds to coincide.…”

“…(almost surely, the specification a.s. is often omitted) and past reconstruction are causally known to both the encoder and the decoder, i.e., S n = X n−1 − a.s. Past treatments of partially observable Gaussian random processes (X n , Y n ) are confined to Gaussian processes such that past reconstruction are causally known to both the encoder and the decoder, i.e., S n = X n−1 [13], [14], [18].…”